Hopefully this will help a few who are still struggling on this, and save you from having to call support or do a reset on your phone...
I have a Verizon Galaxy S4, and for me it turned out to be a queue of auto-updating apps in the background was the issue. I'm assuming the others who cleared their cache had the issue resolved because the background updates were no longer running.
Depending on your settings, you might have auto-updates set to trigger once you are on a wifi connection for example. So in the background there are other applications trying to install or update themselves first, and only until they are all done will the app you want to install get a turn. This would cause it to hang on the "Installing" screen, not the downloading screen.
If your experiencing issues on the "Downloading" part, that would be a network issue with your mobile carrier (or wifi if you have your phone established to an active wifi connection). So to be extra clear, if it's hanging on the "Downloading" part, you may need to call your phone or internet provider for more assistance. If it's hanging on "Installing", then read on...
You can address the "Installing" issue by going into your Google Play settings menu (might look like an icon with three bars next to the Play search menu depending on your phone). From there your phone will present a menu and you will want to look for something called "My Apps". The menu name may be different in future/previous versions of Google Play, so just poke around until you find what you're looking for. In there, you will be able to see if you have a pile of other apps currently attempting to download updates or installs. Either cancel the pending downloads and once again manually install your new app, or just let it go through the updating app queue until it gets to the new app you want to install.
Good luck!